Hodge-Decomposed Gradient Descent on Simplicial Complexes: Self-Referential Optimization via Topological Signal Processing

We introduce Hodge-Decomposed Gradient Descent (HDGD), a novel optimization algorithm for problems defined on simplicial complexes. HDGD exploits the Hodge decomposition of the loss gradient into harmonic, exact, and co-exact components, using each component for a distinct purpose: harmonic components are preserved as structural priors, exact components are discarded as redundant, and co-exact components drive parameter updates. We prove convergence guarantees under standard smoothness assumptions and demonstrate the method on dodecahedral cell complexes, where HDGD discovers geometric parameters matching known physical constants without external supervision. On benchmark graph optimization problems, HDGD achieves comparable convergence rates to Adam while requiring no hyperparameter tuning beyond the complex topology itself. The method establishes a connection between topological data analysis and optimization theory, showing that the Hodge-Laplacian provides a natural decomposition of optimization signals into meaningful geometric categories.